A Morera type theorem for \(L^ 2\) functions in the Heisenberg group
The authors give a Morera type characterization for the square integrable \(CR\) functions on the Heisenberg group. Let \(H^ n\) be the Heisenberg group. For every \(g\in H^ n\), \(\tau_ g(h)=g\circ h\) for \(h\in H^ n\) is the left translation by \(g\), and for a function \(f\) defined on \(H^ n\) \[ \tau_ gf=f\circ\tau_ g \] is the left translation of \(f\) by \(g\in H^ n\). Let \(dV(\alpha,t)\) be the Haar measure on \(H^ n\) \((\alpha\in\mathbb{C}^ n\), \(t\in\mathbb{R})\). Then the main result of the paper is the following Theorem. Let \(\rho_ 1,\rho_ 2,\dots,\rho_ n>0\) and let \(f\in C^ 1(H^ n)\cap L^ 2(H^ n)\). Then \(f\) is a \(CR\) function on \(H^ n\) if and only if \[ \int_{|\alpha|=\rho_ k}\tau_ gf(\alpha,0)\omega_ k(\alpha)=0\quad\text{for every } g\in H^ n\text{ and } k=1,\dots,n, \] where \(\omega_ k(\alpha)=d\alpha_ 1\wedge\cdots\wedge d\alpha_ n\wedge d\overline\alpha_ 1\wedge\cdots\wedge \widehat{d\overline\alpha_ k}\wedge\cdots\wedge d\overline\alpha_ n\).
scientific article; zbMATH DE number 221312